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(True or False) Suppose that $h$ is absolutely integrable on $(a,b)$. If $f$ is continuous on $(a,b)$, if $g$ is continuous and never 0 on $[a,b]$, and if $|f(x)|\leq{h(x)}$ for all $x\in[a,b]$, then $f/g$ is absolutely integrable on $(a,b)$.

My gut tells me this is false, because I can't prove it without making assumptions that weren't given, but I can't think of any counterexamples.

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For a counter example we will need $[a,b]$ to be unbounded i.e. either $a = -\infty$ or $b = +\infty$. Otherwise, the function $f/g$ is continuous on a compact interval; hence attains maximum, minimal on that interval and we always have $f/g$ absolutely integrable because then $|f/g| < M$ for some $M$ so that $\int_{a}^{b} |f/g| \leq M (b - a)$. (There is no need for existence of $h$.)

Given that, take $a = 1, b = +\infty$, $f(x) = 1/x^2$ and $g(x) = 1/x$ and we get a counter example.

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  • $\begingroup$ But $g$ was supposed to be continuous and nonzero on $[a,b\color{red}]$. $\endgroup$ – Hagen von Eitzen Dec 12 '15 at 19:07
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Hint. Let $m=\inf |g|$. Then $(f/g)$ is continuous and $|(f/g)(x)|\le \frac 1m h(x)$.

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